# Video: Finding the Derivative of a Polar Equation

Given a polar curve defined by π = π(ΞΈ), form an expression for the slope of the curve dπ¦/dπ₯ in terms of π and π.

03:37

### Video Transcript

Given the polar curve defined by π is equal to π of π, form an expression for the slope of the curve the derivative of π¦ with respect to π₯ in terms of π and π.

The question gives us a polar curve defined by the equation π is equal to π of π. And itβs asking us to form an expression for the slope of the curve. Thatβs the derivative of π¦ with respect to π₯. And it wants this in terms of π and π. To do this, we recall the chain rule, which tells us that the derivative of π¦ with respect to π₯ is equal to the derivative of π¦ with respect to π multiplied by the derivative of π with respect to π₯. So, this is the first step. This gives us an equation for our slope function dπ¦ by dπ₯.

We now recall our polar equations, π¦ is equal to π sin π and π₯ is equal to π cos π. We recall the question is asking us to write our slope function in terms of only π and π. So, it would be a good idea to rewrite any equations in terms of π and π where possible. Luckily, we can use the substitution π is equal to π of π to rewrite these polar equations.

At this point, we could calculate the derivative of π¦ with respect to π without any problems. However, we canβt directly calculate the derivative of π with respect to π₯, since weβre given π₯ as a function of π. We can get around this by recalling that multiplying by the derivative of π with respect to π₯ is the same as dividing by the derivative of π₯ with respect to π. So, we can find an expression for our slope function dπ¦ by dπ₯ by first calculating dπ¦ by dπ and then dividing this by dπ₯ by dπ. So, to calculate the derivative of π¦ with respect to π, we differentiate π of π multiplied by sin π with respect to π.

We can differentiate this by recalling the product rule, which tells us the derivative of the product of two functions π’ and π£ is equal to π’ prime multiplied by π£ plus π£ prime multiplied by π’. So, we first differentiate our function π of π to get π prime of π and multiply this by sin π. Then, we differentiate sin π, which we recall is cos π. And we multiply this by π of π. We can then do the same to calculate the derivative of π₯ with respect to π. So, thatβs the derivative of π of π multiplied by the cos of π with respect to π.

We differentiate π of π to get π prime of π and then multiply this by the cos of π. And we add onto this the derivative of the cos of π, which we recall is negative sin π. And we multiply this by π of π. Weβre now ready to substitute this into our equation for our slope function. We have that dπ¦ by dπ₯ is equal to dπ¦ by dπ, divided by dπ₯ by dπ. Substituting the equation we got for the derivative of π¦ with respect to π gives us the numerator of π prime of π multiplied by sin π plus π of π multiplied by the cos of π.

And then, substituting the equation we got for the derivative of π₯ with respect to π gives us a denominator of π prime of π multiplied by the cos of π minus π of π multiplied by the sin of π. And so, we have found the formula for our slope function dπ¦ dπ₯ in terms of the function π and π. And itβs fine to just quote this. We donβt need to derive this every time. And so, in conclusion, we have shown that if we are given a polar curve defined by the equation π of π. Then, we can rewrite the slope function the derivative of π¦ with respect to π₯ as π prime of π multiplied by sin π plus π of π multiplied by the cos of π all divided by π prime of π multiplied by the cos of π minus π of π multiplied by the sin of π.